A noncommutative extension of Mahler’s interpolation theorem

. We prove a noncommutative generalisation of Mahler’s theorem on interpolation series, a celebrated result of p -adic analysis. Mahler’s original result states that a function from N to Z is uniformly continuous for the p -adic metric d p if and only if it can be uniformly approximated by polynomial functions. We prove an analogous result for functions from a free monoid A ∗ to a free group F ( B ) , where d p is replaced by the pro- p metric.